www.mech-sci.net/7/31/2016/
doi:10.5194/ms-7-31-2016
© Author(s) 2016. CC Attribution 3.0 License.
Guaranteed detection of the singularities of 3R robotic manipulators
R. Benoit1, N. Delanoue1, S. Lagrange1, and P. Wenger2
1Laboratoire Angevin de Recherche en Ingénierie des Systèmes (LARIS), 62 avenue Notre Dame du Lac, 49000 Angers, France
2Institut de Recherche en Communications et Cybernétique de Nantes (IRCyN), 1 rue la Noë, 44321 Nantes CEDEX 03, France
Correspondence to: R. Benoit (romain.benoit@etud.univ-angers.fr)
Received: 16 January 2015 – Revised: 7 November 2015 – Accepted: 22 December 2015 – Published: 21 January 2016 Abstract. The design of new manipulators requires the knowledge of their kinematic behaviour. Important kinematic properties can be characterized by the determination of certain points of interest. Important points of interest are cusps and nodes, which are special singular points responsible for the non-singular posture changing ability and for the existence of voids in the workspace, respectively. In practice, numerical errors should be properly tackled when calculating these points. This paper proposes an interval analysis based approach for the design of a numerical algorithm that finds enclosures of points of interest in the workspace and joint space of the studied robot. The algorithm is applied on 3R manipulators with mutually orthogonal joint axes. A pre- processing collision detection algorithm is also proposed, allowing, for instance, to check for the accessibility of a manipulator to its points of interest. Through the two proposed complementary algorithms, based on interval analysis, this paper aims to provide a guaranteed way to obtain a broad characterisation of robotic manipulators.
1 Introduction
Algorithms and methods described in this article are applied to the study of a family of robotic manipulators: 3 revolute- jointed manipulators with mutually orthogonal joint axes.
Those manipulators are first studied because they can be re- garded as the positioning structure of a 6R manipulator with a spherical wrist. A main point is that they can be cuspidal, which means that they can change their posture without hav- ing to meet a singularity, as detailed in Baili et al. (2004) and Wenger (2007). It may or may not be the desired behaviour.
To help the reader understand the notion of non singular posture changing that motivate the study of cuspidal manip- ulators, two videos showing, respectively, a non-singular and a singular posture changing trajectory, are proposed along- side the online version of this paper (see Supplement). For a robot with only revolute joint axis, checking that a configu- ration is singular can be done through a geometrical method.
Indeed, a configuration is singular if the end effector is in a revolute joint axis or if the end effector is on a line that cross all of the actuated revolute joint axis (see Baili, 2004).
A cuspidal robot has at least one cusp in a planar cross section of its workspace. On the other hand, the existence of nodes in this section is intimately related to the existence of voids in the robot workspace. Thus, cusps and nodes are important points of interest (Husty et al., 2008). A classifica- tion based on the number of such points can be established (Corvez and Rouillier, 2004; Baili et al., 2004).
Cusp points ans nodes points are named after the local form admitted by the image of the singular set at such point.
Indeed, a cusp point, in the workspace, is a mathematical cusp point for at least one cross section of the image of the singular set. Similarly, in a cross section of the workspace, a node is located at the crossing of two branches of the image of the singular set.
Formally, a node is defined as a workspace point with two singular inverse kinematic solutions (IKS). In a similar fash- ion, a cusp can be defined as a workspace point with three equal singular IKS. These definitions are the one used in Baili et al. (2004), so as to define a formal condition for the
presence of cusps and nodes through a characteristic polyno- mial.
Studying the nature of singular points instead of only iso- lating them to avoid unstable behaviour is relatively recent (Wenger, 2007). However, this approach is quite comple- mentary to the common objective to detect the singular set of a robot, providing useful information on the properties of the robot, particularly for novel design. Methods for detect- ing the singular set include the brute force method of eval- uating numerically the norm of the determinant of the Ja- cobian (det(Df)) and extracting the set of points minimizing this quantity. At the opposite of the spectrum, the equation det(Df)=0 is formally or implicitly solved and numerical solution may be extracted from this resolution. For formally complicated kinematic function, a middle ground is needed in the form of methods returning precise constraints on the singular points. This middle ground usually implies a gen- eral scheme synergistic with interval analysis methods and will be detailed in Sect. 3.
The main point of the algorithm and methods we are de- tailing here is to use Interval Analysis to enclose, in a guaran- teed way, the cusps and nodes in the generator plane section of the manipulator workspace. To find these points, we use two systems of equations, whose roots are joint space points yielding the cusps and nodes. To enclose the roots of those systems of equations, the Interval Newton method is used.
We will verify that, for manipulators with no internal mo- tion, and with some imprecision in their geometric parame- ters, it is possible to find their cusp and node points, with the formerly introduced algorithms.
Complete studies of manipulator families, as done in Baili et al. (2004), allow one to choose a manipulator within a large range of geometric parameters, when a precise behaviour is needed. Alternatively, algorithms presented in this article make it possible to study manipulators with geometric pa- rameters between chosen bounds. It makes them a first step in guaranteeing the behaviour of a manipulator, given its ge- ometric parameters, and the precision affordable for building the actual manipulator.
2 Studied manipulators
The studied manipulators have three unlimited revolute joints. Thus, it is sufficient to restrict the analysis to their last two joints. Since the workspace is symmetric about the first joint axis, it is enough to restrict its analysis to a planar half cross-section in the plane defined byp
x2+y2, z
, that we will identify to x2+y2, z
for computational purposes.
Figure 1 shows the studied manipulator and its geometric parameters. Note that, for a matter of convenience in our al- gorithms, angles βi have been used instead of the standard αi, whereβi=π/2−αi.
We will first consider the same manipulators as in Baili et al. (2004) that is, manipulators with orthogonal rotations
P
Figure 1.Kinematic diagram of a general 3R manipulator with θ1=0.
c1 c4
c2 c3
Joint space
n2 n1
n1' n2'
C1 C2
C3 C4
N1
N2 Workspace
Figure 2.Singular sets in the space of the last two joints (θ2, θ3) and in the cross section
ρ=p
x2+y2, z
of the workspace, for a 3R orthogonal manipulator with parametersd2=1,d3=1.5,d4= 0.7,r2=0.5,r3=0.5.
and no offset along their last joint. With conventions chosen in Fig. 1, these manipulators are defined byβ2=β3=r3= 0.
Figure 2 shows, for an instance of orthogonal 3R ma- nipulator, the singular sets of its kinematic function, in the joint space (θ2, θ3) and the workspace cross section
ρ=p
x2+y2, z
. Figure 2 also illustrate the nature of the cusps and nodes, as the cusps (C1, C2, C3, C4) and the nodes (N1, N2), are pictured in the workspace cross sec- tion. Their inverse kinematic solutions, on the singular set in the joint space, which are respectively (c1, c2, c3, c4) and ({n1,n10},{n2,n20}), are also pictured, as they are the points we are effectively searching in this paper.
We will show that our methodology is able to provide the same results as in Baili et al. (2004). Furthermore, our ap- proach can also be used for manipulators with an offset along their last joint and always returns an exact enclosure of the searched singular joint space points.
3 Application of Interval analysis
3.1 Interval analysis
Interval analysis is a computing method, that operates on in- tervals instead of operating on values. The point of this is mainly for numerical computation because it allows one to guarantee values to be in intervals (see Jaulin et al., 2001;
Moore, 1996) whose bounds can be exactly stored by a com- puter. Interval analysis is a simultaneous computation of numbers and errors.
In this article, boxes will be vectors of intervals. The no- tion of interval can be extended by Cartesian product, so In- terval analysis can be extended to boxes by the use of inclu- sion maps.
Letf be a map. An inclusion map off is a function[f] that associates to a boxD, a box[f](D) such thatf(D)⊂ [f](D). Note that (x∈D⇒f(x)∈ [f](D)).
In practice, the inclusion map[f]off is chosen to mini- mize the boxes[f](D) with respect to inclusion.
This computing method is useful for its usability when a limited set of values can be exactly represented, as for nu- merical computations. In this case, a pointP is represented by the smallest boxDcontainingP andf(P) is represented by [f](D), the smallest box in the image space containing f(D).
3.2 Interval analysis in Robotics
Interval analysis is a tool that, due to its properties seen in Sect. 3.1, can be used for many applications in Robotics (see Merlet, 2011) such as computing the kinematics of manipu- lators, including parallel ones.
One of the robotic applications of Interval Analysis is sin- gularity analysis, that is, finding singular points of the kine- matic map of a manipulator. To find those singular points, a general scheme is used, which consists of a subdivision and shrinking process on the box of study. The main idea is that the searched points are defined as roots of an equation. Then, any box whose image by the map associated with the equa- tion does not contain 0, does not contain any searched point.
If a box may contain a root, then an operator is used to shrink the box to smaller ones containing the roots in the initial box.
Ultimately, when the box cannot be reduced this way, it is cut into several sub-boxes that are studied again. An instance of this scheme, to enclose the singular points of manipula- tors, can be found in Bohigas et al. (2012) and Bohigas et al.
(2013). What makes the general scheme synergistic with In- terval Analysis, is that they both operate on boxes and have the purpose to enclose computed values.
As stated previously, several methods, using Interval Anal- ysis or not, exist to enclose the singular points of a manip- ulator. But, it is also necessary to verify the nature of those singular points. For instance, suppose you succeeded in find- ing the enclosure of the singular set in the workspace as in
Figure 3.Two identical box coverings with two possible couples of covered curves.
Fig. 3. The real singular set can be either one of the two in- stance depicted in this Fig. 3. To conclude on the behaviour of the manipulator, it is necessary to verify if the two curves intersect or not.
In this paper, we propose an algorithm to enclose specific singular points that define the behaviour of a manipulator, us- ing Interval analysis. Accordingly, next subsection proposes a method to enclose numerically roots from a system of equa- tions, through Interval Analysis: the Interval Newton method.
3.3 The Interval Newton algorithm
Given a square system of equations described byf =0, we can define an operator over boxes. This Interval Newton op- eratorNf associated to the mapf is defined by:
Nf :D7−→x−
(df(D))−1×f(x)
, (1)
whereDis a box andx∈D.
df(D) is the matrix of intervals enclosing all the matrices associated to the linear map of the differential off at a point inDand (.)−1is the operator of matrix inversion. In practice, in our algorithm, (.)−1is computed applying the formulae of the inverse of a matrix. It should be noted that, in Eq. (1), instead of (df(D))−1×f(x), any set 6(D, f(x)) could be used, as long as it encloses the solutionsw of Aw=f(x) whereA∈df(D).
The main point is that the topological relation betweenD andNf(D) depends on the presence of a root inD:
1. ifNf(D)⊂Dthen∃!x∈Dsuch asf(x)=0 andx∈Nf(D),
2. ifNf(D)∩D=∅then@x∈Dsuch asf(x)=0, 3. ifNf(D)∩D6=∅then (if∃x∈Dsuch asf(x)=0
thenx∈Nf(D)∩D).
Figure 4.Interval Newton Algorithm.
The Interval Newton method applied with f is defined (see Neumaier, 1990) as being the Algorithm following the flow diagram of Fig. 4.
The Interval Newton algorithm is able to find the roots of a square system of equations if the Jacobian matrix associated with it is invertible for the roots of the studied system, imply- ing that the Interval Newton method can only find isolated roots.
The Interval Newton method can also fail if the chosen precision is not small enough. For instance it can allow a studied box with a size smaller than the precision to contains several roots. One then has to choose a smaller precision, such as no box can contain several roots.
4 Finding cusps and nodes
4.1 Kinematic map and singularity concepts
We should first recall that Cusps points and nodes points in the workspace are singular positions of the end effector sat- isfying some additional properties: a cusp admits three equal inverse kinematic solutions and a node admits two distinct pairs of equal inverse kinematic solutions. Instead of search- ing for those points in the workspace, we are searching for their inverse kinematic solutions. Our points of interest are then what we are defining as Cusps and nodes in the joint
space which are the sets of the singular inverse kinematic so- lutions of the cusp points and node points in the workspace, respectively.
In the case of 3R orthogonal manipulators, using the conventions of Fig. 1, due to their invariance along the rotation of parameter θ1 we consider the joint space JS defined by JS= {(θ2, θ3)| −π≤θ2< π; −π≤θ2< π}. Similarly, instead of considering the entire workspace, we consider a generator cross section of the workspace, SWS, that can be easily defined as the cylinder coordinates around thezaxis, minus the angular coordinate. A position (x,y,z) in the workspace is then converted as (ρ=x2+y2, z). From these consideration, the manipulator kinematic map can be expressed as f =(f1, f2):R2⊃JS7−→SWS=R2 with f1(θ2, θ3)=(cos(θ2) (d4cos(θ3)+d3)+d2−r3sin(θ2))2+ (d4sin(θ3)+r2)2andf2(θ2, θ3)=sin(θ2) (d4cos(θ3)+d3)+ r3cos(θ2).
An internal motion occurs when the end tip point P reaches a joint axis. In this case, the inverse kinematics ad- mits a continuum of solutions, which forms a line in the joint space. On any box that intersect an internal motion line, the proposed algorithm cannot conclude.
4.2 Applying the Interval Newton algorithm
Applying the Interval Newton algorithm to find cusps and nodes requires to define those points and pairs of points as roots of square systems of equations. We will then consider the same properties and characterisation of cusp and nodes points, in the joint space, that were developed in Delanoue and Lagrange (2014). Additionally, the situations where the defining systems are degenerated will be handled in a non- trivial manner to allow a quicker execution of the constructed algorithm.
4.2.1 Application to the cusps
In the following, df refers to the differential off and Df refers to the Jacobian matrix off, which is the matrix asso- ciated to df.
Geometric considerations: we consider that a joint cusp point, C, is a point for which the orthogonal of Ker(df(C)) is collinear with the gradient of the singular curve, defined by det(Df)=0. It is worth noting that inR2, being collinear with a vectorv=(v1;v2)6=0 is the same as being orthogo- nal to the vectorw=(−v2;v1)6=0. Also, if Df(P)6=0, the rows of Df are a base of the orthogonal of Ker(df(P)) and as long as Df(P) is invertible, the orthogonal of Ker(df(P)) is of dimension 2 and thus it cannot be collinear with grad(det(Df))(P). Putting all of this together, we can con- clude that if grad(det(Df))(P) is not the null vector and Df(P)
is not the null matrix, thenP is a cusp point if:
∂f1
∂θ2
(P)·
−∂det(df)
∂θ3
(P)
+∂f1
∂θ3
(P)·∂det(df)
∂θ2
(P)=0
∂f2
∂θ2
(P)·
−∂det(df)
∂θ3
(P)
+∂f2
∂θ3
(P)·∂det(df)
∂θ2
(P)=0 .
(2) Specificities for the algorithm: system (2) is square, which allows one to use the Interval Newton Method to find its iso- lated roots. The roots of system (2) that we are searching are singular points. Then, we will apply the Interval Newton Method only if a studied box contains a singular point, that is, if det(Df) may be null on the box. The final point is that grad(det(Df))(P) and Df(P) must not be null for the searched rootsP, in order to detect those. Then, we will always verify that the components of grad(det(Df)) and Df(P) are not null on the boxes that should contain a cusp-root. If it is not the case on one of the isolated box, it will be cut into pieces that will be studied again.
4.2.2 Application to the nodes
Geometric considerations: node points are much simpler than cusp points for transcription in roots of a map. Indeed, let 1E be the diagonal ofE, that is1E= {(a, a)|a∈E}. Then, we are searching for couples (x1, x2)∈R2×R2−1R2, satisfying:
f(x1) =f(x2) det(Df(x1)) =0 det(Df(x2)) =0
(3)
Specificities for the algorithm: to apply the Interval Newton method to the system (3), this system needs to be a square one, which is the case here, with 4 joint variables and 4 equa- tions. We search the roots in JS×JS⊂R2×R2while avoid- ing the roots in1JS⊂1R2, because on this last subset, the Jacobian matrix associated with the system (3) is not invert- ible while having roots and the Interval Newton method fails.
LetSj be the singular set off (in the joint space JS). In- stead of applying the time consuming process of verifying that a studied box does not intersect 1JS and verifying the injectivity off, restricted to a subset ofSjeach time the in- tersection occurs, one can build a covering ofSj verifying a well chosen property. Indeed, if the covering is done so that any intersecting boxes admit a hull on whichf, restricted to Sj, is injective, then, it suffices to apply Interval Newton al- gorithm with system (3) to couples of disjoint boxes, in this last covering.
Note that the covering, built along with the process, is a guaranteed covering of the singular set.
5 Performances of the Algorithms
5.1 Implementing and running the cusp and node algorithms
All results in this section are valid for any value, or interval of values, ofr3.
To implement, in C++, the algorithms defined in Sect. 4.2, for 3 revolute-jointed manipulators with mutually orthogo- nal joint axes, formal expressions of the derivatives and ma- trices derived fromf, needed in the algorithms, were calcu- lated. The algorithms evaluate the needed expression on the required boxes, replacing the standard functions and oper- ators by corresponding inclusion maps. To handle intervals and operations on them, the library “Filib++” is used.
The application to more general 3 revolute-jointed manip- ulators, withβ26=0 orβ36=0, can be done by calculating their kinematic map. But, as the formal expressions increase in length, the running time of the algorithm may increase and the precision needed to enclose the interest points may need to be higher.
In the implemented algorithms, the initial box of study for (θ2, θ3) can be defined using any box or list of boxes, inR2. The box of geometric parameters can also be chosen. Our algorithms can also be coupled with a procedure enclosing the usable joint space, given a simple volumetric model of the manipulator. The returned enclosure may also be chosen as the boxes of study.
Table 2 shows results returned by the algorithms, ap- plied to examples of classes of studied parameters for 3 revolute-jointed manipulators with orthogonal axes, re- ported in Table 1, and with an initial box of study for (θ2, θ3) of [−3.1415,3.1415] × [−3.1415,3.1415] close to the[−π, π] × [−π, π]full range for the joint angles.
5.2 The cusp enclosing Algorithm
Manipulator inducing no indeterminate (cases a, b, d and e of Table 1): the algorithm has been applied to every example of geometric parameters sets in Baili et al. (2004). When the manipulator does not have an internal motion, for a moderate precision, the algorithm needs little time to find the rigorous enclosures of the cusps, and does not return any indetermi- nate box.
Manipulator inducing indeterminate (case f of Table 1):
when the algorithm is applied to a robot that has internal mo- tions, it finds the cusps outside the internal motions, with the same running time as before. The algorithm then has to run for some time until it encloses the lines associated with the internal motions with boxes whose size is the chosen preci- sion. The running time is then dependant of the chosen pre- cision.
Table 1.Some studied cases of robotic manipulators.
Characteristics Geometric parameters Properties of manipulator
Designation d2 d3 d4 r2 r3 Internal motion Cusps Nodes
a 1 2 1.5 1 0 no 4 0
b 1 2 1.5 1 0.5 no 4 0
c [1, 1.001] [2, 2.001] [1.5, 1.501] [1, 1.001] 0 NA 4 NA
d 1 [0.7] [0.3] [0.2] 0 no 0 0
e 1 1.5 [0.7] 0.5 0 no 4 2
f 1 0.5 [1.3] [0.2] 0 yes 0 2
Table 2.Algorithms performances on the robotic manipulators of Table 1.
Cusp algorithm Node algorithm
Case Precision Cusps Indeterminate Time Precision Nodes Indeterminate Time Improved time
a 10−4 4 no 32 s 2.5×10−10 0 no 10 h 23 min
b 10−4 4 no 46 s 2.5×10−10 0 no 18 h 45 min
c 10−4 4 no 35 s 2.5×10−10 NA yes NA out of memory
d 10−4 0 no 12 s 2.5×10−10 0 no 52 s 16 s
e 10−4 4 no 52 s 2.5×10−10 2 no 35 h 5 h and 42 min
f 10−2 0 yes 12 min 10−2 2 yes 42 s 42 s
f 10−3 0 yes 90 min 10−3 2 yes 41 s 41 s
The running times are given for a computer with a 64 bits operating system and an Intel®Core™i7 CPU.
When the parameterpis not computer storable, then it is replaced by the smallest interval containing it, noted[p].
5.3 The nodes enclosing Algorithm
On boxes where there is no cusps and no internal motion lines (cased of Table 1) the nodes enclosing algorithm concludes after a running time close to the one needed for the cusp en- closing algorithm with no internal motion. However, when the box includes a cusp (cases a,b and eof Table 1) the running time of the algorithm increases quite significantly, because, near cusps, f restricted toSj, is injective only on small boxes. In the same way, the Interval Newton method can conclude, only on small boxes when the hull box of its two components is close to a cusp point.
5.3.1 Performance improvement using contraction methods
As it has been formerly noted, the main drawback of the algorithm is its relatively slow check of the absence of nodes near cusps. To improve on this, we decided to rely on the contraction method library Ibex, available freely at http://www.ibex-lib.org/, with documentation.
A Contractor is an operator on Boxes, associated to a set, that reduce the box to a smaller box without removing any el- ement of the associated set. Contraction methods are used in Interval Analysis to enclose a set. It relies on contractors, as- sociated to the chosen set, and may use subdivisions, so as to get a enclosure of the chosen set. The main interest of those
methods is that reducing a box using contractors is a lot less time consuming than bisecting it until a chosen precision.
An Ibex contraction procedure is included in the algorithm as an additional check before applying an iteration of the node Interval Newton method on a couple of disjoint boxes.
The procedure is based upon a contractor using the Interval Newton method with the system dedicated to the node as pa- rameter. As the Ibex procedure’s contractor reduce quite ef- ficiently the studied boxes, we use it as a quick way to check the absence of node in a couple of boxes (see as a box of double dimension). If the procedure return an empty box as a result, then, there is no node in the initial couple of boxes and it is not needed to apply any subdivision process or interval Newton iterations further.
As a result of including the Ibex calling test in the node searching step, the performances of the algorithm toward the length of checking the absence of nodes have been greatly improved. For instance, the time needed to execute the node searching step, for a 3R manipulators with nodes and cusps is decreased to less than a fifth of its value (caseeof Table 1).
5.4 Application with boxes of geometric parameters Our algorithms have been implemented to handle intervals of geometric parameters, so to use intervals of parameters (as for casebof Table 1) it is only needed to define a box of geometric parameters which is not restricted to a point.
+
Figure 5.Oblong solid model as a Minkowski sum of a segment and a ball.
If the algorithms find a solution box, then, for any set of geometric parameter in the defined box of parameters, there is a single interest point in the solution box. There will be no interest point in any box that is neither a solution box nor an indeterminate box for any set of geometric parameter, in the defined box of parameters. Ultimately, it can exist interest points, for any set of geometric parameter in the defined box of parameters, only in solution boxes and in indeterminate boxes. For a manipulator with an internal motion, the algo- rithms return, at least, enclosures for a subset of the interest point and a covering of the research space that can contain interest points.
6 Collisions detection through Interval Analysis An complementary procedure have been added to our algo- rithm, allowing the user to get an enclosure of the set of pa- rameters inducing collisions and of the set of parameters in- ducing no collisions at all.
6.1 Used model
Solids that may collide (either elements of the manipulator’s kinematic chain or environment obstacle) are considered ob- long object defined by a segment and a radius, where the ob- long object is the set of all points distant to the segment from at most the defining radius, see Fig. 5. With this model, two objects collide if and only the distance between the respec- tive defining segments is equal or less than the sum of the two defining radius.
6.2 Implemented procedure
The implemented procedure is based upon the SIVIA in- version algorithm, and consists in applying it for the distance between every pair of defined segments. As it implies com- puting the minimum of the distance between a point in one segment and a point in a second one, the two segments are split until a limit size and the distance between each couple of sub-segment is checked if greater than the sum of the ra- dius.
As the distance varies with the articular parameters, the former process is applied for sub-boxes of the initial list of boxes of articular parameters. To sum up, the procedure is
Algorithm 1 Set Inversion Via Interval Analysis (SIVIA) al- gorithm
Require: A setS, a function, and a real number(a limit of size) and a list of boxes of researchL
return 3 lists of boxesI,OandU whileLis not empty do
extractBfromL
evaluateD=f(B) through Interval Analysis ifD=f(B)⊂Sthen
addBtoI
else ifD=f(B)∩S=∅then addBtoO
else ifsize(B)> then
splitBinB1andB2and add them toL else
addBi toU end if end while
In the end (∪B∈IB)⊂S⊂(∪B∈(I∪U)B) andS∩(∪B∈OB)=
∅
applying a list of consecutive double-SIVIA for each couple of solids that may collide, the user defined to be studied.
6.3 Joint use with cusp and node detection
The interest of the collision detection procedure in itself is to control the feasibility of given paths in the joint space, by the studied manipulators. Combined with the the knowl- edge of an enclosure of the singular set, returned by the pre- processing step of the node enclosing procedure, one can also check for the possibility to join two posture by a non singular feasible path.
The joint use of the procedure with the main detection al- gorithm also allows, quite naturally, to check for the manip- ulator access to the chosen interest points. In our case, the inaccessibility to the cusps and nodes may not change the behaviour of the manipulator. However, for instance, a wide collision zone around a cusp, may imply, that the manipula- tor is, in practice, not cuspidal if no articular path can link two IKS without crossing a singularity or inducing collision.
7 Conclusions
The main interest of the proposed method is that it can be used to find any isolated point of interest for the evaluation of the behaviour of any manipulator, provided it can be defined by a root of a square system of equations. Then, this method- ology constitutes a possible way of describing a robotic ma- nipulator singular set, allowing for the guaranteed detection of isolated specific singular points of interest.
It is to be noted that most of the running time of the algo- rithm is used to treat boxes where the Interval Newton algo- rithm fails to conclude. To increase the performance of the
algorithm, alternate methods for splitting and localized tests need to be used and are still searched.
As for a lot of Interval Analysis algorithms, our algorithm can be time consuming when dealing with complicated kine- matic functions or high dimension boxes of study, especially for the nodes enclosing algorithm, due to the doubled di- mension of the box of study, although attenuated by a pre- subdividing in the joint space. However, provided that the al- gorithm runs for the time needed with a sufficient precision, it is able to find enclosures for the searched points without errors, or at least a subset of those enclosures and a covering of the searched points.
With the joint use of the collision procedure, the algo- rithm aims to provide efficient and guaranteed information about the manipulator’s kinematic properties. The algorithm could provide additional information that may be relevant to the user’s interests with additional procedures to, for in- stance, enclose the singular positions in the workspace or enclose the non-singular IKS of the singular positions. As such, the reader may found the source code to the algo- rithms at http://perso-laris.univ-angers.fr/~benoitr/contenu/
thom_2d_online.zip.
The Supplement related to this article is available online at doi:10.5194/ms-7-31-2016-supplement.
Edited by: D. Pisla
Reviewed by: two anonymous referees
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